anshw5 - 1.4.7 Finish the following proof for Theorem...

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1.4.7 . Finish the following proof for Theorem 1.4.12. Assume B is a countable set. Thus, there exists f : N ! B , which is and onto. Let A B B . We must show that A is countable. Let n 1 = min f n 2 N : f ( n ) 2 A g . g : N ! A , set g (1) = f ( n 1 ) . Show how to inductively continue this process to produce a function g from N onto A . Shouldn±t one say, ²Show how to continue this process inductively³? Notice that f n 2 N : f ( n ) 2 A g is f 1 ( A ) , so we can write n 1 = min f 1 ( A ) . Then write n 2 = min n 2 f 1 ( A ) : n > n 1 ± n 3 = min n 2 f 1 ( A ) : n > n 2 ± and, in general, n i +1 = min n 2 f 1 ( A ) : n > n i ± That±s the construction. Because A is in´nite, and f maps onto B , the set of natural numbers n 2 f 1 ( A ) : n > n i ± is nonempty no matter what the natural number n i is. So this set has a minimum element. Clearly n 1 < n 2 < n 3 < ±±± by the way we chose them. We de´ne

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This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.

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anshw5 - 1.4.7 Finish the following proof for Theorem...

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